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104
Poisson Surface Reconstruction
, 2006
"... We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function ..."
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Cited by 369 (5 self)
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We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.
Stability of Persistence Diagrams
, 2005
"... The persistence diagram of a realvalued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result ..."
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Cited by 222 (23 self)
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The persistence diagram of a realvalued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.
ExampleBased 3D Scan Completion
 EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING
, 2005
"... Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete sur ..."
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Cited by 85 (23 self)
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Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete surface scans, using a database of 3D shapes to provide geometric priors for regions of missing data. Our method retrieves suitable context models from the database, warps the retrieved models to conform with the input data, and consistently blends the warped models to obtain the final consolidated 3D shape. We define a shape matching penalty function and corresponding optimization scheme for computing the nonrigid alignment of the context models with the input data. This allows a quantitative evaluation and comparison of the quality of the shape extrapolation provided by each model. Our algorithms are explicitly designed to accommodate uncertain data and can thus be applied directly to raw scanner output. We show on a variety of real data sets how consistent models can be obtained from highly incomplete input. The information gained during the shape completion process can be utilized for future scans, thus continuously simplifying the creation of complex 3D models.
Spectral Surface Reconstruction from Noisy Point Clouds
, 2004
"... We introduce a noiseresistant algorithm for reconstructing a watertight surface from point cloud data. It forms a
..."
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Cited by 81 (1 self)
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We introduce a noiseresistant algorithm for reconstructing a watertight surface from point cloud data. It forms a
Consolidation of Unorganized Point Clouds for Surface Reconstruction
"... We consolidate an unorganized point cloud with noise, outliers, nonuniformities, and in particular interference between closeby surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optim ..."
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Cited by 47 (10 self)
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We consolidate an unorganized point cloud with noise, outliers, nonuniformities, and in particular interference between closeby surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised, outlierfree and evenly distributed particles over the original dense point cloud, so as to improve the reliability of local PCA for initial estimate of normals. Next, an iterative framework for robust normal estimation is introduced, where a prioritydriven normal propagation scheme based on a new priority measure and an orientationaware PCA work complementarily and iteratively to consolidate particle normals. The priority setting is reinforced with front stopping at thin surface features and normal flipping to enable robust handling of the closeby surface sheet problem. We demonstrate how a point cloud that is wellconsolidated by our method steers conventional surface generation schemes towards a proper interpretation of the input data. 1
Provably good sampling and meshing of surfaces
 Graphical Models
, 2005
"... The notion of εsample, introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an εsample of a C2continuous surface S for a sufficiently small ε, then the Delaunay triangulation of E restricted to S is a goo ..."
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Cited by 37 (9 self)
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The notion of εsample, introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an εsample of a C2continuous surface S for a sufficiently small ε, then the Delaunay triangulation of E restricted to S is a good approximation of S, both in a topological and in a geometric sense. Hence, if one can construct an εsample, one also gets a good approximation of the surface. Moreover, correct reconstruction is ensured by various algorithms. In this paper, we introduce the notion of loose εsample. We show that the set of loose εsamples contains and is asymptotically identical to the set of εsamples. The main advantage of loose εsamples over εsamples is that they are easier to check and to construct. We also present a simple algorithm that constructs provably good surface samples and meshes. Given a C2continuous surface S without boundary, the algorithm generates a sparse εsample E and at the same time a triangulated surface DelS(E). The triangulated surface has the same topological type as S, is close to S for the Hausdorff distance and can provide good approximations of normals, areas and curvatures. A notable feature of the algorithm is that the surface needs only to be known through an oracle that, given a line segment, detects whether the segment intersects the surface and, in the affirmative, returns the intersection points. This makes the algorithm useful in a wide variety of contexts and for a large class of surfaces. Keywords: Surface mesh generation, εsampling, surface approximation, restricted Delaunay triangulation, mesh refinement
Interactive decal compositing with discrete exponential maps
 ACM Trans. Graph
, 2006
"... Figure 1: A clay elephant statue (left) was modeled using sketchbased implicitsurface modeling software. Then, a lapped base texture and 25 feature textures were extracted from 22 images taken with a digital camera and composited on the surface. Photography, image creation, and texture positioning ..."
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Cited by 34 (7 self)
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Figure 1: A clay elephant statue (left) was modeled using sketchbased implicitsurface modeling software. Then, a lapped base texture and 25 feature textures were extracted from 22 images taken with a digital camera and composited on the surface. Photography, image creation, and texture positioning was completed in under an hour. A method is described for texturing surfaces using decals, images placed on the surface using local parameterizations. Decal parameterizations are generated with a novel O(N logN) discrete approximation to the exponential map which requires only a single additional step in Dijkstra’s graphdistance algorithm. Decals are dynamically composited in an interface that addresses many limitations of previous work. Tools for image processing, deformation/featurematching, and vector graphics are implemented using direct surface interaction. Exponential map decals can contain holes and can also be combined with conformal parameterization to reduce distortion. The exponential map approximation can be computed on any point set, including meshes and sampled implicit surfaces, and is relatively stable under resampling. The decals stick to the surface as it is interactively deformed, allowing the texture to be preserved even if the surface changes topology. These properties make exponential map decals a suitable approach for texturing animated implicit surfaces.
Interactive topologyaware surface reconstruction
 ACM Trans. Graph
"... www.cs.tau.ac.il/˜{asharf,shagil,stoledo,dcor}. www.mat.pucrio.br/˜tomlew. Abstract. The reconstruction of a complete watertight model from scan data is still a difficult process. In particular, since scanned data is often incomplete, the reconstruction of the expected shape is an illposed proble ..."
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Cited by 30 (8 self)
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www.cs.tau.ac.il/˜{asharf,shagil,stoledo,dcor}. www.mat.pucrio.br/˜tomlew. Abstract. The reconstruction of a complete watertight model from scan data is still a difficult process. In particular, since scanned data is often incomplete, the reconstruction of the expected shape is an illposed problem. Techniques that reconstruct poorlysampled areas without any user intervention fail in many cases to faithfully reconstruct the topology of the model. The method that we introduce in this paper is topologyaware: it uses minimal user input to make correct decisions at regions where the topology of the model cannot be automatically induced with a reasonable degree of confidence. We first construct a continuous function over a threedimensional domain. This function is constructed by minimizing a penalty function combining the data points, user constraints, and a regularization term. The optimization problem is formulated in a meshindependent manner, and mapped onto a specific mesh using the finiteelement method. The zero levelset of this function is a first approximation of the reconstructed surface. At complex undersampled regions, the constraints might be insufficient. Hence, we analyze the local topological stability of the zero levelset to detect weak regions of the surface. These regions are suggested to the user for adding local inside/outside constraints by merely scribbling over a 2D tablet. Each new user constraint modifies the minimization problem, which is solved incrementally. The process is repeated, converging to a topologystable reconstruction. Reconstructions of models acquired by a structuredlight scanner with a small number of scribbles demonstrate the effectiveness of the method.
Topology guaranteeing manifold reconstruction using distance function to noisy data, Research Report 429 (2005), available at http://math.ubourgogne.fr/topo/chazal/publications.htm
"... Given a smooth compact codimension one submanifold S of R k and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisyapproximation that g ..."
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Cited by 22 (4 self)
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Given a smooth compact codimension one submanifold S of R k and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisyapproximation that generalize sampling conditions introduced by Amenta & al. and Dey & al. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. Our results allow to consider balls of different radii. For the first approximation condition, we also prove that a subset (known as the λmedial axis) of the medial axis of R k \ K is homotopy equivalent to the medial axis of S. We obtain similar results for smooth compact submanifolds S of R k of any codimension.